A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs

State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable

The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems (TDSs) have been found, including similar pattern formation and propagation of localized structures. An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolutions in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between 1D SESs and TDSs. Real-world systems in physics, chemistry, biology, economy, etc. are typically described by a large number of equations, involving many variables, and therefore, their dynamical evolution occurs in a high dimensional phase space. One of the most exciting discoveries in the field of dynamical systems in the last decades is that, in spite of their high dimensionality, these systems can be described by low-dimensional attractors, which can be reconstructed even if one can only observe one variable, during a finite time interval, with finite resolution and with large measurement noise. Examples of such high dimensional systems are one-dimensional spatially extended systems (1D SESs), and time delayed systems (TDSs). In a space-time representation, these systems show similar phenomena (e.g., wave propagation, pattern formation, defects and dislocations, turbulence, etc.). In this work we study the state space reconstruction of these systems, from the time series of one scalar “observed” variable. We analyze a bistable reaction-diffusion 1D SES and two TDSs: a bistable scalar system with delayed feedback, and a system composed by two lasers with delayed mutual cross coupling (the system has several variables and two time-delay terms). We find that their dynamics can be reconstructed in a three-dimensional pseudo-space, where the evolution is governed by the same polynomial potential.Peer ReviewedPostprint (published version

Realization of Pseudo-Rational Input/Output Maps and Its Spectral Properties

Realization of a special class of input/output maps is considered. A linear input/output map which belongs to this class is called pseudo-rational, and it behaves in a way similar to the input/output maps of finite-dimensional systems in the following sense : To determine the canonical state space, only the output data on a bounded time interval is needed. Central examples of this class of input/output maps are those of delay-differential systems. A concrete representation for the canonical space is given ; and then it is used to give an explicit differential equation description. Some spectral properties which are very similar to those of the delay-differential systems are also proved. Some examples are demonstrated to illustrate the realization procedure

Exit time asymptotics for small noise stochastic delay differential equations

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE) with coefficients that depend on the history of the process over a finite delay interval. We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for a broad class of infinite-dimensional small noise stochastic equations.Comment: 39 page

Capacity of Control for Stochastic Dynamical Systems Perturbed by Mixed Fractional Brownian Motion with Delay in Control

In this paper, we discuss the relationships between capacity of control in entropy theory and intrinsic properties in control theory for a class of finite dimensional stochastic dynamical systems described by a linear stochastic differential equations driven by mixed fractional Brownian motion with delay in control. Stochastic dynamical systems can be described as an information channel between the space of control signals and the state space. We study this control to state information capacity of this channel in continuous time. We turned out that, the capacity of control depends on the time of final state in dynamical systems. By using the analysis and representation of fractional Gaussian process, the closed form of continuous optimal control law is derived. The reached optimal control law maximizes the mutual information between control signals and future state over a finite time horizon. The results obtained here are motivated by control to state information capacity for linear systems in both types deterministic and stochastic models that are widely used to understand information flows in wireless network information theory. The contribution of this paper is that we propose some new relationships between control theory and entropy theoretic properties of stochastic dynamical systems with delay in control. Finally, we present an example that serve to illustrate the relationships between capacity of control and intrinsic properties in control theory.Comment: 17 pages, 2 example

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs